Pensieve header: $\rho_1$ for up to 12 crossings for Radmila.

SetDirectory["C:\\drorbn\\AcademicPensieve\\People\\Sazdanovic"];

Once[<<KnotTheory`];

Rot::usage="Rot[K] where K is any n-crossing knot presentation returns {Cs, φ}, where Cs is a length n list of crossings as triples {s,i,j} and φ is a lengtth 2n list of rotation numbers.";

PD[epd_EPD]:=PD@@epd/.{(Subscript[X, i_,j_]):>X[j,i+1,j+1,i],(Subscript[(OverBar[X]), i_,j_]):>X[j,i,j+1,i+1]}

Rot[pd_PD]:=Module[{n,xs,x,rots,Xp,Xm,front={1},k},
n=Length@pd;rots=Table[0,{2n}];
xs=Cases[pd,x_X:>(Piecewise[{{Xp[x[[4]],x[[1]]], PositiveQ@x}, {Xm[x[[2]],x[[1]]], True}}])];
For[k=1,k<=2n,++k,
If[FreeQ[front,-k],
front=Flatten@Replace[front,k->(xs/.{
Xp[k,l_]|Xm[l_,k]:>{l+1,k+1,-l},
Xp[l_,k]|Xm[k,l_]:>(++rots[[l]];{-l,k+1,l+1}),
_Xp|_Xm:>{}
}), {1}],
Cases[front,k|-k]/.{k,-k}:>--rots[[k]];
]
];
{xs/.{Xp[i_,j_]:>{+1,i,j}, Xm[i_,j_]:>{-1,i,j}},rots} ];
Rot[K_]:=Rot[PD[K]];

The Program

(Subscript[R, 1])[s_,i_,j_]:=s((Subscript[g, j,i])((Subscript[g, (SuperPlus[j]),j])+(Subscript[g, j,(SuperPlus[j])])-(Subscript[g, i,j]))-(Subscript[g, i,i])((Subscript[g, j,(SuperPlus[j])])-1)-1/2);
Z[K_]:=Module[{Cs,φ,n,A,s,i,j,k,Δ,G,ρ1},
{Cs,φ}=Rot[K]; n=Length[Cs];
A=IdentityMatrix[2n+1];
Cases[Cs,{s_,i_,j_}:>(A[[{i,j},{i+1,j+1}]]+=(({{-(T^s), (T^s)-1}, {0, -1}})))];
Δ=(T^((-Total[φ]-Total[Cs[[All,1]]])/2))Det[A];
G=Inverse[A];
ρ1=(∑_(k=1)^n)(Subscript[R, 1])@@Cs[[k]] - (∑_(k=1)^(2n))φ[[k]]((Subscript[g, k,k])-1/2);
Factor@{Δ,(Δ^2)ρ1/.(SuperPlus[α_]):>α+1/.(Subscript[g, α_,β_]):>G[[α,β]]}];

The First Few Knots

TableForm[Table[Join[{(Subscript[K[[1]], K[[2]]])},Z[K]], {K,AllKnots[{3,6}]}], TableAlignments->Center]

Monitor[
tab=Table[Join[{K},Z[K]], {K,AllKnots[{3,12}]}],
K]

tab>>"rho1to12xings.m"